m1 internship report

simulation of single photonemission in dot-in-rod nanocrystal

Laboratoire Kastler Brossel, May-July 2014

Author: Zou junwen, Supervisor: Alberto bramati, Quentin glorieux

Introduction

Laboratoire Kastler Brossel

The Kastler Brossel Laboratory (LKB) is one of the main actor in the field of fun-damental physics of quantum systems. The laboratory was funded in 1951 by AlfredKastler and Jean Brossel. This lab is attached to three agencies including the NationalCentre for Scientific Research(CNRS), the Ecole Normale Superieure(ENS) and the Uni-versite Pierre et Marie Curie(UPMC). Many new themes have appeared recently in thisfield, such as quantum entanglement or Bose-Einstein Condensation in gases, whichleads to a constant renewal of the research carried out in the laboratory.Presently its activity focus on several themes: cold atoms (bosonic and fermionicssystems), atom lasers, quantum fluids, atoms in solid helium, quantum optics, cavityquantum electrodynamics, quantum information and quantum theory of measurement,quantum chaos, high-precision measurements. The themes lead not only to a better un-derstanding of fundamental phenomena, but also to important applications, like moreprecise atomic clocks, improvement of interferometric gravitational wave detectors andnew methods for biomedical imaging[1].

Quantum Optics

The group research area is centered on the quantum properties of light that is pro-duced by many different optical systems. It consists of experimental and theoreticalstudies concerning the shaping of quantum fluctuations of light, the generation of quan-tum correlations and entangled states, the interaction between quantum light and mattereither dilute or condensed, and the use of non-classical light to improve the sensitivityof optical measurements. The groups is mostly interested in the so-called ”continuousvariable regime”, in which one considers the quantum properties of the electric field ofthe optical wave, and where single photons are not distinguishable.The rapidly growing field of quantum information requires in particular specific resourcesto operate its protocols. The group puts lots of efforts in the realization of quantumdevices such as non-classical light sources, quantum memories and quantum repeaters.In this very competitive area, the group has proposed and developed many original con-tributions.Recently, the group has extended the study of quantum effects in optical systems tothe more general thematics of multimode quantum optics where the system under studyhas many degrees of freedom and thus possibly conveys a huge amount of information.This is for example the case of optical images, which span over many transverse modes,and of optical frequency combs which span over many frequency modes. The groupexplores this subject both theoretically and experimentally, from the introduction ofnew concepts to the improvement of highly sensitive measurement and applications toquantum information processing.The group is also interested in interaction between quantum optics and condensed mat-ter: polariton quantum gas in semi-conductor quantum well microcavities, generationof single and entangled photons by semiconductor nanocrystals.[2]

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1 Basic Concept

1.1 Exciton

An exciton is a bound state of an electron and an electron hole which are attractedto each other by the electrostatic coulomb force. The exciton, an electrically neutralquasiparticle, exists in insulators, semiconductors and in some liquids. It can transportenergy without transporting net electric charge, so it is regarded as an elementaryexcitation in the condensed matter[3].

1.2 Biexciton

A biexciton is created from two free excitons in the condensed matter. In quan-tum information and computation, it is essential to construct coherent combinations ofquantum states. We can use the basic quantum operations to perform the physicallydistinguishable quantum bits. Therefore, biexcitons can be illustrated by a simple four-level system.

In an optically driven system which is shown on Figure 1, the |01y and |10y statescan be directly excited by |00y state. However, direct excitation of the upper |11y statefrom the ground state is usually forbidden. In order to get the biexciton state, the mostefficient alternative is coherent nondegenerate two-photon excitation. In other words, itis used |01y or |10y state as an intermediate state.

|01y and |10y states are the cases of the linearly polarized (i.e. the symmetric andantisymmetric combinations of spin up and down) exciton states. Their energy is dif-ferent due to the different combination of spin. Moreover, due to coulomb coupling, thebiexciton state |11y state is combined with the two antiparallel-spin excitons[4].

Figure 1: Model of single Quantum Dot. There are 4 states: ground state:|00y, directexcited state: |01y and |10y, upper excited state: |11y. The selection rule is shown bythe arrow.[5]

1.3 Quantum dot

Quantum dots are artificial atoms that can be custom designed for a variety of ap-plications. Specifically, it is made of semiconductor materials which are small enoughto exist quantum mechanical properties. However, quantum dots have some man-made

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nanoscale structures in which electrons can be confined in all 3 dimensions. Due to itsstructure, its electronic properties are intermediate between those of bulk semiconduc-tors and of discrete molecules[6].

1.4 Photo-luminescence

Photoluminescence is the phenomenon of luminescence from direct photoexcitationthe emitting species. Specially, its light emits from any form of matter after the ab-sorption of photons (electromagnetic radiation). According to the quantum theory, thisprocess can be treated as the transition from the high energy state to the low energystate. In the experiment measure, the absorption of photons is provided by the laser[7].

1.5 Colloidal core/shell nanocrystal

Colloidal core/shell nanocrystals(NCs) contain at least two semiconductor materialsin an onionlike structure. The core nanocrystals have lot of basic optical properties,for example, their fluorescence wavelength, quantum yield and lifetime[8]. Particularly,wet-chemically synthesized colloidal core/shell nanocrystals will emit non classical lightat room temperature, which makes them suitable for the application of the quantumoptics[9].

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2 Experimental measurement

M.Manceau etc [9] has already done the experiment of the CdSe/CdS core-shell dot-in-rods(DRs) synthesis and photoluminescence measurement in the lab. In their pa-per, they have measured the flickering between the bright state and grey state of thenanocrystal and showed that the autocorrelation function can be identified the excitonand biexciton state.

2.1 Exciton Experiment

In the experiment, a fast switching(flicking) between two states of emission(a brightstate with a high emission efficiency and a grey state with a lower emission efficiency isshown by the Figure 2.

Figure 2: (a)PL intensity, (b)histogram of PL intensity, (c)normalized ACF and (d)PLdecay in the dot-in-rod sample 1(DR1)[9].

2.1.1 PL intensity

In their experiment, the photoluminescence(PL) was collected by a confocal micro-scope with a high numerical aperture oil immersion objective. In the Figure 2a, thePL intensity is counted in second(x-axis) and the intensity counting is done with thebin time 250µs. Specially, the zoom of the PL intensity Figure 2a can show that therewill be two different types of light: red one belongs to the bright state and green onebelongs to the grey state. However, the blank zone between two states are disorder,which implies that the mechanism between the switch of two states is not clear.

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2.1.2 Histogram of PL intensity

In order to know the transition between the bright and grey states, histogram of thePL intensity will be used when fitting the data. In the Figure 2b, a two poissoniandistribution fit is used to know the mean emission intensity of the grey/bright state.Two peaks appear in the histogram and the data histogram curve doesn’t coincide thepoissonian fitting curve.

2.1.3 PL decay

In the Figure 2d, the whole PL intensity will be separated into two curves(Brightstate and grey state) depending on the intensity count number in Figure 2a.

2.1.4 Autocorrelation function(ACF)

As a fast switching between a bright and a grey mode with biexcitons, histogramof the PL intensity is not enough to describe the detail of the transition between twostates. In order to characterize PL photons emission from the DRs, Time-CorrelatedSingle Photon Counting(TCSPC) is used in the experiment, which is corresponding tothe Figure 2c. In the experiment, due to the too fast flicking e.g. switching between twostates is less than 100 ms, TCSPC can’t detect two photons in the same time, whichimplies that the PL beam have to be separated into two beams.

Figure 3: Normalized ACF for DR at short timescales. a. Grey state Normalized ACF.b. Bright state Normalized ACF. c. Whole PL normalized ACF.[9]

Autocorrelation function is used to know the detail of the photons inforamtion withtwo different time instants t1and t2 . It can be measured by the second order autocorre-lation function of the electric field intensity which is related to the beam. It is defined

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as

gp2qpt1, t2q �xE�pt1qE

�pt2qEpt1qEpt2qy

x|Ept1q|2y x|Ept2q|2y�

xIipt1qIipt2qy

xIipt1qy xIipt2qy(1)

where ¡ represents the statistic average value and E is the electrostatic field and I isthe emission intensity of each state.

In the experiment, autocorrelation function in the classical case is written as:

gp2qpτq �xI1ptqI2pt� τqy

xI1ptqy xI2pt� τqy(2)

where τ is the delay time between two channels of TCSPC and Iiptq (i=1, 2)is theemission intensity of bright and grey state in the experiment.

In the quantum autocorrelation function, it will be:

gp2qpτq �xS1ptqS2pt� τqy

xS1ptqy xS2pt� τqy(3)

where Siptq is the time of each photon and τ is the delay time.When we consider the delay between the photon with its neighbor photons, quantum

autocorrelation will be used, for example, the normalized autocorrelation function withshort time delay in Figure 3. When we consider the classical autocorrelation betweenthe intensity event of the histogram, its classical autocorrelation function can be writtenfunction 2. Its figure is Figure 2c in several delays between the photons.

3 Simulation Model of photons with Excitons

In order to describe the mechanism of the transition between bright and grey state,the new model of photoluminescence(PL) will be simulated in this internship report.From the Figure 2, it is proved that there will be two states in the PL emission. In thismodel, there will be two different kinds of photons with two kinds of emission microtime,two kinds of time distributions of the photon, two probabilities of detection by differentAPD(avalanche photodiode) with two macrotimes.Note: 1) Microtime is the time of certain PL photon emission after the correspondinglaser photon exciting. Its unit is ns. 2) Macrotime is the time of certain PL photonemission after the first laser photon exciting the whole system. Its unit is µs.

3.1 Exponential statistics coefficient τA and τB

The time of PL photons emitted from the nanocrystal will follow the exponentialstatistics, which is shown by F.Pisanello[10]. In the semiconductor CdSe/CdS DRs,there exists two states(bright state and grey state), which implies that two differentexponential statistics coefficient will fit the photon emission time.

The equations of the emission time for the bright state with exponential statisticscoefficient τA and the grey state with another exponential statistics coefficient τB areshown below:

P pδtAq � e�δtAτA (4)

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Figure 4: Exciton photon detection in microtime

P pδtBq � e�δtBτB (5)

where δtA{B is the microtime of the PL photon and P is the PL emission time of eachstate.

3.2 Power law statistics coefficients γA, γB and their thresholdvalues mA, mB

The switching of the emission photons between the grey mode and the bright modewill follow the power law statistics[11]. In the bright mode, γA is the power law coeffi-cient and mA is its threshold value. In the grey mode, γB represents the coefficient ofthe power law in the grey state and mB is its corresponding threshold value.Note: Threshold value means the minimum number of photons series in the power lawstatistics.

When considering the bright state with the power law coefficient γA and the greystate with the power law coefficient γB, their corresponding switch emission series areshown below:

P pTAq �1

T γA(6)

P pTBq �1

T γB(7)

where γ is the coefficient of the power law statistics and P is the PL emission time.

3.3 Normal distribution coefficients σA or σB in each state

When the power law statistics is considered as the transition between the bright andgrey mode, its threshold value will have changed. In order to control the change ofthe threshold value, normal distribution will be considered. In this model, 2 kinds ofnormal distribution are designed to control the change of the threshold values in the

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corresponding power law statistics:

P pxq � e�pmi�m�

i q2{2σ2

(8)

where m�i (i=A,B) is the mean value of the threshold in the power law distribution and

the σi (i=A,B) is the standard deviation.

When the bright state is considered, its threshold value has mean value m�A and the

normal distribution coefficient σA. The grey state has the similar normal distributioncoefficient σB and mean value of threshold m�

B.

3.4 Probabilities coefficients PA and PB

As the limitation of the avalanche photodiode, especially for double photons emittedby the biexciton excitation, some photons can’t be detected. Therefore, the probabilitiesof the detection PA and PB will be added into the exciton simulation part. Specifically,PA is the probability of the bright state photons which can be detected by (channel 1 ofAPD)APD1 and PB is the probability of the grey state photons which can be detectedby APD2(channel 2 of APD).

3.5 Precision of the detection

The model of photons with excitons has to be considered the measurement accuracyby the instruments. The photon emission time has smallest value, which is treated asthe precision in this model.

3.6 Pulse time of the laser excitation 4tIn the real experiment, the laser with pulses will be used. The interval of each pulse

of the laser will be treat as 4t.

3.7 Number of Photon

In this model, it will simulate from one to ten million photons system.

4 Model of Biexciton

As the structure of biexciton, sometimes the nanocrystal will emit two photons after apulse laser excitation. Model of biexciton is designed to describe this case. In this model,four extra coefficients will be added into the simulation. Cconsidering the biexcitonstate of the bright state, an exponential statistics coefficient τ1 and its correspondingprobability P1 will be utilized. They has the similar equation as the exciton part 3.1and 3.4. An biexciton of the grey state also has its exponential statistics coefficient τ2and probability P2.Note: In the model of biexciton, the probabilities P2, P1 of the biexcition and theprobability of exciton PA, PB are independent.

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Figure 5: Biexciton photon detection in microtime

4.1 Exponential statistics coefficient τ1 and τ2 of biexciton

Biexciton sometimes will emit 2 photons when laser pumping the nanocrystal. Thephoton emission time of biexciton also follows the exponential statistics. Accurately,the photon emitted by the bright state biexciton has an exponential statistics coefficientτ1 and the photon emitted by the grey state biexciton has an exponential statisticscoefficient τ2In detail,

P pδt1q � e�δt1τ1 (9)

P pδt2q � e�δt2τ2 (10)

where δt1{2 is the microtime of the PL photon and P is the PL emission time of eachstate.

4.2 Probabilities of photon emitted by biexciton in the detec-tor

In the real experiment, not all photons will be detected by the APD. In order toimprove the precision of the Biexciton Model, probabilities P1 is considered the brightstate and P2 is for the grey state. Moreover, the bright state emission has only P1

probability data from the APD1, the similar probability P2 for the grey by APD2.

5 Simulation

In the simulation part, Matlab will be used as the programming Language. All theMatlab codes are in the appendix.

5.1 Simulation of exponential statistics

When simulating the PL time, the inverse function method is used to build the expo-nential statistics.Original function:

fpxq � e�tτ (11)

Inverse function:t � �τ lnpfpxqq (12)

where f(x) is the uniform distribution between 0 and 1.

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5.2 Simulation of power law statistics

The power law statistics is also built through the inverse function.Mathematical Method:Original function:

P pT q �1

T γ(13)

where the equation has the minimum value ’m’ of the PL emission time P .

x � mp1 � rq�1γ (14)

where m is the threshold value of the power law distribution, γ is the coefficient of thepower law distribution and the r is the uniform distribution between 0 and 1[12].

5.3 Simulation of PL intensity figure

In the Matlab, Step to build the figure: 1) Use the histogram bin count order ”histc”to count the number of values in each bin time that are within each specified bin range.The bin time can be different as it is set by the function. 2) Build the histogram throughthe order ”plot”.

5.4 Simulation of Quantum correlation function(ACF)

Step to build the figure of quantum correlation function: 1) Separate the photonemission time into two APDs with the same probability 50%. 2) Build two same sizeof emission time vector of the bright and the grey state. 3) Use the element translationand subtraction of the vector to get the delay of the each photon. 4) Calculate the ACFand plot the figure.

5.5 Simulation of Classical correlation function(ACF)

Step to build the figure of classical correlation function: 1) Calculate the PL intensityvalue as the section 5.3. 2) Build two same size of time vector of the intensity. 3) Usethe element translation and subtraction to build the ACF function. 4) Calculate theACF values with different delay time.

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6 Result

Exciton

In this case, I simulate the photoluminescence(PL) emission excited by exciton fromthe DRs and compare the result with the experiment data in Figure 2.

Table 1: Simulation data of photon emitted by the exciton of dot-in-rod nanocrystal

τA τB PA PB γA γB mA mB σA σB step(ns) 4t(µs) num60 10 0.4 0.13 2 2 5 3 10 16 0.4 512 106

Figure 6: PL intensity of the DR (dot-in-rod) emission (left) and Histogram of PLintensity of the DR (right)

In these two figures(Figure 2a, 2b and Figure 6), there all exist two types of emissiontime photons. Compared with the position of peaks, both cases have two peaks at thealmost same intensity(x-axis). However, the switch between the bright and grey stateis different in the simulation part. Specifically, the number of event drops a little in theexperiment figure, but the switch zone of two state has a concave plane in the simulation.The reason of the difference is that double photons (biexciton) sometimes will emit inthe experiment. To some extent, the model of exciton can describe the mechanism ofthe transition.

In the Figure 2c, the autocorrelation function in the bottom increases a little at ashort time scale and then decreases until the value of 1. In the simulation Figure 7, gp2q

factor has decreased from about 1.3 to 1.1 at the large delay time. The most differencebetween 2 cases is in the range of 10�2. When delay time is large, both cases gp2q factorwill decrease to the value of near 1 finally.

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Figure 7: gp2q of the PL intensity emission from the excitons on the certain delay betweenphotons

Biexciton

In Table 2, four extra terms will be added into simulation (τ1 , τ2 , P1 and P2 ). Inthis simulation, two photons will be emitted at the same time with their probabilitiesP1 and P2 due to the biexciton.

Table 2: Simulation data in the Biexciton model

τA τB PA PB τ1 τ2 P1 P2 γA γB mA mB σA σB step(ns) 4t(µs) num60 10 0.08 0.5 2 2 0.02 0.02 2 2 25 1 5 2 512 0.4 107

Compared with the Figure 8a, The PL intensity Histogram in the simulation has thesimilar intensity distribution. The ratio between the bright state and the grey state isabout 7.5. In the Figure 8a, the ratio between the bright and grey state is about 7.3. Inthe switching zone of two states, the simulating curve has better fit with the experimentcurve in Figure 8a compared with the Model of exciton.

When we consider the classical correlation in the Figure 10(Right), we can observethe gp2q is about 1.4 at the short delay times and it will turn to 1.05 at a much largerdelay time. In the quantum autocorrelation function, the gp2qp0q has ratio about 0.16compared with the highest value of ACF in Figure 10(Left). On the whole, gp2qp0q inthe simulation part fits well compared with the experiment measurement the Figure 8d.

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Figure 8: (a) Histogram of the PL intensity of DR2 whose Count rates below 30counts/ms in green are associated with the grey state, while the part of the histogramabove 100 counts/ms is attributed to the bright state. (b) PL decay curves for greystate photons (green circles) and bright state photons (red squares). (c) g(2) for DR2on several decades of delays between the photons. From top to bottom: g(2) for the greystate photons, bright state photons together with the g(2) of the whole PL intensity.(d) ACF for DR2 at short time scales. From top to bottom: ACF for the grey statephotons, bright state photons together with the ACF of the whole PL intensity[9].Note: DR2 is the second type of dot-in-rod nanocrystal in the experiment

Figure 9: Histogram of the PL intensity in the biexciton simulation

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Figure 10: PL quantum autocorrelation for the DRs at short time scales(Left) and theclassical correlation gp2q factor for the biexciton(Right)

6.1 Examples of the coefficient effect in the Model

Effect of the bintime in the histogram

Table 3: exciton in different bintime

τA τB PA PB γA γB mA mB σA σB step(ns) 4t(µs) num bintime(µs) label60 10 0.4 0.13 2 2 1 3 10 6 0.4 512 106 100 160 10 0.4 0.13 2 2 1 3 10 6 0.4 512 106 300 260 10 0.4 0.13 2 2 1 3 10 6 0.4 512 106 500 360 10 0.4 0.13 2 2 1 3 10 6 0.4 512 106 700 460 10 0.4 0.13 2 2 1 3 10 6 0.4 512 106 1000 5

From Figure 11, it is easily to find that when the bin time increases, the peak willleft move with the lower intensity peak. The transition area thebetween the grey andbright mode will appear a small peak when the bin time becomes larger. In order toknow accurately the effect of the bin time, 3-D curve is plotted in Figure 12.

In Figure 12(Left), the tendency of the grey mode peak will increase with the increas-ing bin time, and the opposite intensity tendency for the bright mode. At the top viewof the 3D curve(Figure 12(Right)), it is easy to find that two peaks will be separatedincreasingly with the growing bin time. The distribution of each peak will be broadenedwhen the bin time increases.

Effect of the different probabilities in the biexciton emission time

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Figure 11: Biexciton PL intensity with different bin time.They are upper left 100µs,upper right 500µs, middle 300µs, lower left 700 µs and lower right 1000µs

Figure 12: 3D plot of the exciton(Left) and Low view of the curve(Right)

Table 4: biexciton in different probabilities

τA τB PA PB τ1 τ2 P1 P2 γA γB mA mB σA σB step(ns) 4t(µs) num label60 10 0.13 0.4 1.9 2 0.05 0.05 2 2 5 2 15 2 512 0.4 10e7 160 10 0.13 0.4 1.9 2 0.2 0.2 2 2 5 2 15 2 512 0.4 10e7 260 10 0.13 0.4 1.9 2 0.3 0.3 2 2 5 2 15 2 512 0.4 10e7 3

Figure 13: quantum correlation function with different probabilities in the biexcitonemission.PA and PB for each figure: 0.05(Left), 0.2(Middle), 0.3(Right)

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From the Figure 13, we can observe that gp2qp0q becomes larger with the increasingprobability of the biexciton. When we increase the count density of the PL emission, thevalue of the gp2qp0q will change a little until the count intensity is very large. Moreover,gp2qp0q will trend a certain value at all cases. The probabilities of the biexciton will alsochange the trend of the gp2q in the non-zero delay.

Effect of the normal distribution factor in the biexciton emission time

In the switch of the two states, the normal distribution factor of the threshold valuewill have an influence on the distribution of the intensity for each state. From the Table5, the distribution of the threshold value in state 1 is changed.

Table 5: biexciton in different normal distribution factor

τ1 τ2 P1 P2 τA τB PA PB γ1 γ2 m1 m2 σ1 σ2 step(ns) 4t(µs) num label60 10 0.13 0.4 1.9 2 0.05 0.05 2 2 5 2 15 2 512 0.4 107 160 10 0.13 0.4 1.9 2 0.05 0.05 2 2 5 2 10 2 512 0.4 107 260 10 0.13 0.4 1.9 2 0.05 0.05 2 2 5 2 5 2 512 0.4 107 3

Figure 14: PL intensity with different threshold value distribution. Normal distributionfactor σ1 for each figure: 15(Left), 10(Middle), 5(Right)

From the Figure 14, we can find that the bright mode intensity increases when thenormal distribution factor σ1 decreases, which means that the normal distribution factorcan affect the height of the intensity peak. And the position of two peaks can’t beaffected by the change of the normal distribution.

7 Conclusion

In this report, it is successfully to simulate the photoluminescence emission time ofthe bright and grey mode. The simulation models predict the different cases of thetransition between the bright and grey mode. Through these cases, it can be found outthe mechanism of the photoluminescence in the dot-in-rod nanocrystal.

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References

[1] Antoine Heidmann. Kastler brossel laboratory. http://www.lkb.ens.fr/

Kastler-Brossel-Laboratory.

[2] Quantum optics. http://www.lkb.ens.fr/-Optique-Quantique,29-.

[3] Exciton. http://en.wikipedia.org/wiki/Exciton.

[4] Bixciton. http://en.wikipedia.org/wiki/Biexciton.

[5] Gang Chen, TH Stievater, ET Batteh, Xiaoqin Li, DG Steel, D Gammon,DS Katzer, D Park, and LJ Sham. Biexciton quantum coherence in a single quan-tum dot. Physical review letters, 88(11):117901, 2002.

[6] Muhammad Usman. Quantitative Modeling and Simulation of Quantum Dots. PhDthesis, Purdue University, Apr 2011.

[7] Alan D McNaught and Alan D McNaught. Compendium of chemical terminology,volume 1669. Blackwell Science Oxford, 1997.

[8] Peter Reiss, Myriam Protiere, and Liang Li. Core/shell semiconductor nanocrystals.small, 5(2):154–168, 2009.

[9] M. Manceau, S. Vezzoli, Q. Glorieux, F. Pisanello, E. Giacobino, L. Carbone,M. De Vittorio, and A. Bramati. Effect of charging on cdse/cds dot-in-rods single-photon emission. Phys. Rev. B, 90:035311, Jul 2014.

[10] Ferruccio Pisanello, Godefroy Lemenager, Luigi Martiradonna, Luigi Carbone, Ste-fano Vezzoli, Pascal Desfonds, Pantaleo Davide Cozzoli, Jean-Pierre Hermier, Elis-abeth Giacobino, Roberto Cingolani, et al. Non-blinking single-photon generationwith anisotropic colloidal nanocrystals: Towards room-temperature, efficient, col-loidal quantum sources. Advanced Materials, 25(14):1974–1980, 2013.

[11] M Kuno, DP Fromm, HF Hamann, A Gallagher, and DJ Nesbitt. On/off flu-orescence intermittency of single semiconductor quantum dots. The Journal ofChemical Physics, 115(2):1028–1040, 2001.

[12] Aaron Clauset, Cosma Rohilla Shalizi, and Mark EJ Newman. Power-law distribu-tions in empirical data. SIAM review, 51(4):661–703, 2009.

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Appendices

Appendix A Exciton Code

1 %input2 %tau1 is the rate parameter of the exponential random statistics ...

for the state A(bright mode). Its unit is ns.3 %tau2 is the rate parameter of the exponential random statistics ...

for the state B(grey mode). Its unit is ns.4

5 %P1 is the probability of detection in state A6 %P2 is the probability of detection in state B7

8 %Gamma1 is the power law random number coefficient for state A9 %Gamma2 is the power law random number coefficient for state B

10 %thre1 is the mean value of threshold in the state A power law ...statistics

11 %thre2 is the mean value of threshold in the state B power law ...statistics

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13 %sigmal1 is the normal distribution of the threshold for state A14 %sigmal2 is the normal distribution of the threshold for state B15

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17 %deltatime is the laser excitation time unit(us)18 %step is the precision for each detector unit(ns)19 %sizeOut is the total number of the photon20

21 %output22 %mac is the real photon emission time in us(macrotime)23 %lmac is the label for each macrotime24

25 %method: Inverse function of the distribution26 % origional function:f(x)=exp(t/Tau)27 % f(x) inverse function is t=-mu*ln(f(x)); and if f(x) distributes28 % evenly,the t is the exponential random distribution .29

30 % precision setting31 % tau=step*number of step32 % we can get the number of step first: mu.step=tau/step33 % we get the intergal of step and then we multiply with the step ...

and get34 % the tau value whose minimum value is the step except35

36 % state switch method: random vector for only 0 and 1 values37 % we use power law statistics to bulid a random number of 0 and 1.38

39

40 function [mac,lmac]= ...exciton(tau1,tau2,P1,P2,Gamma1,Gamma2,thre1,thre2,sigma1,sigma2,deltatime,step,sizeOut)

41

42 % set the parameters use in the function43 l1 = powerlaw(thre1,thre2,sigma1,sigma2,Gamma1,Gamma2,sizeOut); % ...

product random number between 0 and 1 for certain percentage of ...

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state 144 l2 = 1-l1; % product random number between 0 and 1 for ...

state 245

46 step=step/1000; %step in ps47

48 mu1step=tau1/step; %get the number of step in tau 149 mu2step=tau2/step; %get the number of step in tau 250

51 stepval1=round(mu1step);52 stepval2=round(mu2step);53

54 b = rand(1,sizeOut);55 b1 = round(stepval1.*(log(b)));56 b2 = round(stepval2.*(log(b)));57

58 r1 = -step.*b1; %bulid the exponential random number for tau 159 r2 = -step.*b2; %bulid the exponential random number for tau 260

61 %bulid the probability of emission62 p1=rand(1,sizeOut);63 p2=rand(1,sizeOut);64

65 p1=p1<P1;66 p2=p2<P2;67

68 r1=r1.*p1;69 r2=r2.*p2;70

71

72 %bulid a mixing exponential random number73 r1 = l1.* r1; %select the exponential random number for tau 174 r2 = l2.* r2; %select the exponential random number for tau 275

76 r = (r1+r2); %combine all the selection random number in ns77 l = l1+l2*2; %find the corresponding states for each exponential ...

random number78

79 %macrotime random number in us80 w=r>0;81 mac=1:sizeOut;82 mac=deltatime*mac+r/1000; %unit of mac is us83

84 %delete the missing photon in the series85 mac=w.*mac;86 mac(mac==0)=[];87 lmac=l.*w;88 lmac(lmac==0)=[];89

90 end

19

Appendix B Bieciton Code

1 %input2 %tau1 is the rate parameter of the exponential random statistics ...

for the state A exciton(bright mode). Its unit is ns.3 %tau2 is the rate parameter of the exponential random statistics ...

for the state B exciton(grey mode). Its unit is ns.4

5 %P1 is the probability of detection in state A exciton6 %P2 is the probability of detection in state B exciton7

8 %tauA is the rate parameter of the exponential random statistics ...for the state A biexciton(bright mode). Its unit is ns.

9 %tauB is the rate parameter of the exponential random statistics ...for the state B biexciton(grey mode). Its unit is ns.

10

11 %PA is the probability of detection in state A biexciton12 %PB is the probability of detection in state B biexciton13

14 %Gamma1 is the power law random number coefficient for state A15 %Gamma2 is the power law random number coefficient for state B16 %thre1 is the mean value of threshold in the state A power law ...

statistics17 %thre2 is the mean value of threshold in the state B power law ...

statistics18

19 %sigmal1 is the normal distribution of the threshold for state A20 %sigmal2 is the normal distribution of the threshold for state B21

22

23 %deltatime is the laser excitation time unit(us)24 %step is the precision for each detector unit(ns)25 %sizeOut is the total number of the photon26

27 %output28 %mac is the real photon emission time in us(macrotime)29 %lmac is the label for each macrotime30 %mac1 is the real photon emission time series of state A31 %mac2 is the real photon emission time series of state B32

33 %method: Inverse function of the distribution34 % origional function:f(x)=exp(t/Tau)35 % f(x) inverse function is t=-mu*ln(f(x)); and if f(x) distributes36 % evenly,the t is the exponential random distribution .37

38 % precision setting39 % tau=step*number of step40 % we can get the number of step first: mu.step=tau/step41 % we get the intergal of step and then we multiply with the step ...

and get42 % the tau value whose minimum value is the step except43

44 % state switch method: random vector for only 0 and 1 values45 % we use power law random number function to bulid a random number ...

of 0 and 1.

20

46

47 % power law statistics is used following method:48 % f(x)=threshold*(1-r)ˆ(-1/(gamma-1)) where r is the random statistics49 % between 0 and 150

51 function [mac,lmac,mac1,mac2]= ...biexciton(tau1,tau2,P1,P2,tauA,tauB,PA,PB,Gamma1,Gamma2,thre1,thre2,sigma1,sigma2,deltatime,step,sizeOut)

52

53 % set the parameters use in the function54 l1 = powerlaw(thre1,thre2,sigma1,sigma2,Gamma1,Gamma2,sizeOut); % ...

product random number between 0 and 1 for certain percentage of ...state 1

55 l2 = 1-l1; % product random number between 0 and 1 for ...state 2

56

57 step=step/1000; %step in ps58

59 mu1step=tau1/step; %get the number of step in tau 160 mu2step=tau2/step; %get the number of step in tau 261 muAstep=tauA/step; %get the number of step in tau A62 muBstep=tauB/step; %get the number of step in tau B63

64

65 stepval1=round(mu1step);66 stepval2=round(mu2step);67 stepvalA=round(muAstep);68 stepvalB=round(muBstep);69

70 b = rand(1,sizeOut);71 a = rand(1,sizeOut);72 b1 = round(stepval1.*log(b));73 b2 = round(stepval2.*log(b));74 bA = round(stepvalA.*log(a));75 bB = round(stepvalB.*log(a));76

77

78 r1 = -step.*b1; %bulid the exponential random number for tau 179 r2 = -step.*b2; %bulid the exponential random number for tau 280 rA = -step.*bA; %bulid the exponential random number for tau A81 rB = -step.*bB; %bulid the exponential random number for tau B82

83 %bulid the probability of emission for biexciton time84 p1=rand(1,sizeOut);85 p2=rand(1,sizeOut);86 p3=rand(1,sizeOut);87 p4=rand(1,sizeOut);88

89 % set the probability of detection90 p1=p1<P1;91 p2=p2<P2;92 p3=p3<PA;93 p4=p4<PB;94

95 r1=r1.*p1;96 r2=r2.*p2;97 rA=rA.*p3;98 rB=rB.*p4;

21

99

100 %bulid a mixing exponential random number101

102 r1 = l1.* r1; %select the exponential random number for tau 1103 r2 = l2.* r2; %select the exponential random number for tau 2104

105 r = (r1+r2); %combine all the selection random number in ns106 l = l1+l2*2; %find the corresponding states for each exponential ...

random number107

108 %bulid a excition mixing random number109 rA = l1.* rA; %select the exponential random number for tau A110 rB = l2.* rB; %select the exponential random number for tau B111

112 rr = (rA+rB); %combine all the selection random number in ns113 ll = l1+l2*2; %find the corresponding states for each exponential ...

random number114

115 %macrotime random number in us116 w1=r>0;117 macw=1:sizeOut;118 mac=deltatime*macw+r/1000; %unit of mac is us119

120 %bulid the biexcition121 w2=rr>0;122

123 mac1=deltatime*macw+rr/1000; %unit of mac1 is us124 mac1=w2.*mac1;125 lmac1=ll.*w2;126

127 mac2=w1.*mac;128 lmac2=l.*w1;129

130 %delete the missing photon in the series131 mac=[mac1;mac2];132 mac=reshape(mac,1,[]);133 mac(mac==0)=[];134

135 lmac=[lmac1;lmac2];136 lmac=reshape(lmac,1,[]);137 lmac(lmac==0)=[];138

139

140 end

22

Appendix C Power Law Statistics Code

1 %function pl is to bulid a random number of 0 and 1 which follows 2 ...powerlaw

2 %distribution series3 %output l is the random number series of 0 and 14

5

6 function [series]=powerlaw(thre1,thre2,sigma1,sigma2,Gamma1,Gamma2,num)7

8

9 %bulid 2 normal distribution for threshold values10 thre11=normrnd(thre1,sigma1,1,num);11 thre22=normrnd(thre2,sigma2,1,num);12

13 %bulid 2 random number between 0 and 114 n1=rand(1,num);15 n2=rand(1,num);16 %bulid 2 power law random number series17 rn1=thre11.*(1-n1).ˆ(-1/(Gamma1-1));18 rn2=thre22.*(1-n2).ˆ(-1/(Gamma2-1));19

20 %get the intergal and absolute series21 rad1=abs(round(rn1));22 rad2=abs(round(rn2));23

24

25 %get a series to describe the number of subvector e.g bulid a ...series like[rad1(1),rad2(1),rad1(2),rad2(2),`````]

26 l = [rad1;rad2];27 l = reshape(l,1,[]);28

29 % delete the extra value in the vector to increase the running ...efficient

30 temp=cumsum(l);31 n=sum(temp<num);32 l(n+2:end)=[];33

34 %limit the l vector values in order to35

36 %%random number 0 and 1 series37 %%e.g l=[1 2 3 4], then RR=[1 0 0 1 1 1 0 0 0 0]38

39 R = arrayfun(@(x) mod(x,2)*ones(1,l(x)),1:length(l),'un',false);40 series = cell2mat(R);41 series(num+1:end)=[];42

43 end

23

Appendix D Classical Correlation Code

1 %function classicalcorrelation is to bulid the classical ...correlation in

2 %exciton photon emission time series3 %input4 %mac is the emission time for exciton5 %output6 %g2 is the classical autocorrelation function7

8 function [g2] = classicalcorrelation(mac)9

10 %find the max value for mac11 w=max(mac);12

13 %bulid the time scale14 l=0:2:w; %count in 2us for the hole photon time series15

16 %count the intensity of every 2us17 bin1=histc(mac,l);18

19

20 g21=zeros(1,100);%bulid a 0 vector21

22 %find the classical autocorrelation function with delay time in 2us.Its23 %range is from 1 to 200us delay time.24 for i=1:10025

26 bin2=bin1;27 bin3=bin1;28 bin2(end-i+1:end)=[];%delete the area which is not calculated29 bin3(1:i)=[];30 g21(i)=mean(bin2.*bin3)/mean(bin2)/mean(bin3); %g2 ...

=mean(I(t)*I(t+tau))/mean(I(t))/mean(I(t+tau))31 end32

33 %bulid another 0 vector34 g22=zeros(1,81);35 l=1;36 %count the autocorrelation with 10us. Its range is from 200us to37 %2000us(2ms)38 for j=100:5:100039 bin2=bin1;40 bin3=bin1;41 bin2(end-j+1:end)=[];42 bin3(1:j)=[];43 g22(l)=mean(bin2.*bin3)/mean(bin2)/mean(bin3); %g2 ...

=mean(I(t)*I(t+tau))/mean(I(t))/mean(I(t+tau))44 l=l+1;45 end46

47

48 %combine 2 vectors to build a whole classical correlation time ...vector for

49 %the histogram y-axis

24

50 g2=[g21,g22];51 %build the delay histogram x-axis52 i=(1:100)*2;53 j=(100:5:1000)*2;54 k=[i,j];55 %plot the figure of the classical autocorrelation function56 figure(1)57 plot(k,g2,'o');58 xlabel('Delay(us)')59 ylabel('g(2)')60

61 end

25

Appendix E Quantum Correlation Code

1 %function quantumcorrelation is to bulid the histogram of quantum ...autocorrelation

2 %input3 %mac1 is the photon emission time series of the state A4 %mac2 is the photon emission time series of the state B5 %output6 %output is the autocorrelation value between APD1 and APD2 ...

detection photon7 %delay time8

9 function [output]=quantumcorrelation(mac1,mac2)10

11 %bulid 1 and 2 value random vector the probability of each state is 50%12 len=rand(1,length(mac1));13 len1=len>0.5;14 len2=1-len1;15

16 %bulid the detection data for each APD(2 cases)17 w11=len1.*mac1; %detection for APD1 exciton of the bright mode18 w12=len2.*mac2; %detection for APD2 biexciton of the grey mode19

20 w1=[w11;w12]; %combine the detection21 w1=reshape(w1,1,[]); %delete the undetection photon emission time22 w1(w1==0)=[];23

24 w21=len1.*mac2; %detection for APD1 exciton of the grey mode25 w22=len2.*mac1; %detection for APD2 biexction of the bright mode26

27 w2=[w21;w22];28 w2=reshape(w2,1,[]);29 w2(w2==0)=[];30

31 %make 2 vector same shape in order to use the vector calculation32 if length(w1)>length(w2)33 w1(length(w2)+1:end)=[];34 else35 w2(length(w1)+1:end)=[];36 end37 %bulid a blanck vector38 output=[];39 %precions: 21 elements around delay time40

41 for k=-10:1:1042 %method: element shift for one vector and use another vector minus ...

it to43 %get the delay time44

45 w3=w2(mod((1:end)-k-1,end)+1)-w1;46 w4=w1(mod((1:end)-k-1,end)+1)-w2;47 output=[output,w3,w4];48 end49 %make the matrix as 1*n vector50 output=reshape(output,1,[]);

26

51 output(abs(output)>5)=[];52 end

27

Appendix F The Code Of PL Curve

1 %mac is the series of photon emission time in us2 %l is the counting number per (bintime)us3 %rangecount is the range of counting in s4 %k1 is the intensity count(counts/bintime(us))5 %numevent is the number of event6

7

8 function [numevent,rangecount]=plotcurve(mac,bintime)9

10

11 %find the max value for num12 w=max(mac);13 %bulid the time scale14 %set the bin time15 l=0:bintime:w; %count in us16

17 %count the intensity18 rangecount=histc(mac,l);19

20 l1=l/1000000;%x axis in s21 %plot the pl intensity on seconds22 figure(1)23 plot(l1,rangecount);24 xlabel('time(s)')25 ylabel('count/bintime(us)')26

27

28 %intensity count in unit 529 k1=0:5:max(rangecount)+10;30 %number of events31 numevent=zeros(1,length(k1));32

33 for w=1:1:length(k1)34 numevent(w)=sum((rangecount>5*(w-1)&rangecount<w*5).*rangecount);35 end36 %plot the intensity and the number of event figure37 figure(2)38 plot(k1,numevent,'--o');39 xlabel('count intensity(count/bintime)');40 ylabel('number of events')41

42

43 end

28

Appendix G The Change Of Bin Time

1 %bulid the 3d curve of the intensity count with different bintime2 %inensitycount is the intensity count vector3 %bin is the bin time vector4 %x is the number of the counting for the certain count intensity5 %num is the input photon emission time6 function [intensitycount,bin,x]= bintime(num)7 %bulid a blank vector for the number of counting8 x=[];9

10

11 %bulid the matrix of the intensity and count number with different ...bin time

12 for bintime=100:100:100013

14 %find the max value for num15 w=max(num);16 %bulid the time scale17 %set the bin time18 l=0:bintime:w; %count in us19

20 %count the intensity21 k=histc(num,l);22

23

24 %find the max value of the count intensity25 l1=0:1000:w;26 intensitycount=histc(num,l1);27

28

29 %intensity count in unit 530 intensitycount=0:5:max(intensitycount)+10;31 %number of events vector32 su=zeros(1,length(intensitycount));33

34 for w=1:1:length(intensitycount)35 su(w)=sum((k>5*(w-1)&k<w*5).*k);36 end37

38 %put each numbers of events into a matrix39 x =[x;su];40 end41

42 %plot 3d curve43 bin=100:100:1000;44

45 surf(intensitycount,bin,x);46 xlabel('Intensity(Count/bintime)');47 ylabel('bintime')48 zlabel('number of event')49

50 end

29